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  • Integrate the functions in Exercises 1 to 24. 3

Integrate the functions in Exercises 1 to 24.

    Q3.    \frac{1}{x\sqrt{ax-x^2}}    [Hint: Put x = \frac{a}{t}]

Answers (1)

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Let

                                       x = \frac{a}{t}\ dx\ \Rightarrow \ dx\ =\ \frac{-a}{t^2}dh

Using the above substitution we can write the integral is                                    

                                \int \frac{1}{x\sqrt{ax-x^2}}\ =\ \int \frac{1}{\frac{a}{t}\sqrt{a.\frac{a}{t}\ -\ (\frac{a}{t})^2}} \frac{-a}{t^2}dt

or                                                               

                                                      =\ \frac{-1}{a}\int \frac{1}{\sqrt{(t-1)}}dt

or                                

                                                      =\ \frac{-1}{a}\ (2\sqrt{t-1})\ +\ C

or                                                  =\ \frac{-1}{a}\ (2\sqrt{\frac{a}{x}\ -\ 1})\ +\ C

or                                                   =\ \frac{-2}{a}\ \sqrt{\frac{a-x}{x}}\ +\ C

Posted by

Devendra Khairwa

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